A text book's proof
I think it's very easy to prove the following rule $A\cup (B\cap C)$ easier way than our text book!
The proof is here:
\[A\cup (B\cap C)\]
\[= \{x:x\in A\text{ or }x\in (B\cap C)\}\]
\[= \{x:x\in A\text{ or }(x\in B\text{ and }x\in C)\}\]
\[= \{x:(x\in A\text{ or }x\in B)\text{ and }(x\in A or x\in C)\}\]
\[= \{x:x\in A\cup B\text{ and }x\in A\cup C\}\]
\[= \{x:x\in (A\cup B)\cap(A\cup C)\}\]
\[=(A\cup B)\cap (A\cup C)\]
\[[proved]\]
so, why our text book contains so hard proof?
The proof is here:
\[A\cup (B\cap C)\]
\[= \{x:x\in A\text{ or }x\in (B\cap C)\}\]
\[= \{x:x\in A\text{ or }(x\in B\text{ and }x\in C)\}\]
\[= \{x:(x\in A\text{ or }x\in B)\text{ and }(x\in A or x\in C)\}\]
\[= \{x:x\in A\cup B\text{ and }x\in A\cup C\}\]
\[= \{x:x\in (A\cup B)\cap(A\cup C)\}\]
\[=(A\cup B)\cap (A\cup C)\]
\[[proved]\]
so, why our text book contains so hard proof?
Last edited by Hasib on Wed Dec 15, 2010 6:23 pm, edited 7 times in total.
A man is not finished when he's defeated, he's finished when he quits.
Re: A text book's proof
Please, Any one can say me why the space doesn't working in upper equetions??
A man is not finished when he's defeated, he's finished when he quits.
Re: A text book's proof
you have to indicate spacing if you want to put spaces inside equation.
Form http://www.artofproblemsolving.com/Wiki ... TeX:Layout
For example x \in A\text{ or } x\in B gives $x \in A\text{ or } x\in B$
Form http://www.artofproblemsolving.com/Wiki ... TeX:Layout
BTW if you want to write text inside equation you better use \text{your text}However, if you do need to tweak the spacing in math mode, there are some special commands:
\, a small space
\: a medium space
\; a large space
\quad a really large space
\qquad a huge space
\! a negative space (moves things back to the left)
Here are examples of these in action:
For example x \in A\text{ or } x\in B gives $x \in A\text{ or } x\in B$
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Re: A text book's proof
Moon wrote:you have to indicate spacing if you want to put spaces inside equation.
Form http://www.artofproblemsolving.com/Wiki ... TeX:LayoutBTW if you want to write text inside equation you better use \text{your text}However, if you do need to tweak the spacing in math mode, there are some special commands:
\, a small space
\: a medium space
\; a large space
\quad a really large space
\qquad a huge space
\! a negative space (moves things back to the left)
Here are examples of these in action:
For example x \in A\text{ or } x\in B gives $x \in A\text{ or } x\in B$
thanks vaia a lot!!!
A man is not finished when he's defeated, he's finished when he quits.
Re: A text book's proof
বইয়ের প্রমাণ আর তোমার প্রমানে তো তেমন পার্থক্য পাইলাম না। বইয়ে জিনিসটা বিপরীত দিক থেকেও করা আছে, যেটা করা উচিৎ।hasib.mo wrote:I think it's very easy to prove the following rule $A\cup (B\cap C)$ easier way than our text book!
The proof is here:
...
btw, এখানে কিছু latex code দিলাম যেটা মনে হয় তোমার কাজে লাগবেঃ
\Rightarrow = $\Rightarrow$
\Leftrightarrow = $\Leftrightarrow$
Every logical solution to a problem has its own beauty.
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Re: A text book's proof
ও আচ্ছা, বইয়ে এইটা সরাসরি প্রমাণ নাই। $A\cup(B\cup C) = (A\cup B)\cup (A\cup C)$ এইভাবে প্রমান করে লিখছে তোমার করা প্রমানটা একইভাবে করা যায়।
Every logical solution to a problem has its own beauty.
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Re: A text book's proof
Ha. The text book contains the proof with two side, and they use the method of subset($\subset$) bt, i use that logic bt, in a changed form(i use directly equal sign!). So, i think i don't need to do the proof in two side with the changed form. thanks!
Bt, { } (bracket) isn't working in the dollar sign! What will i do?! And thanks again!
Bt, { } (bracket) isn't working in the dollar sign! What will i do?! And thanks again!
A man is not finished when he's defeated, he's finished when he quits.
Re: A text book's proof
bracket is a part of LaTeX command...so you have to use \{x\} for $\{x\}$
A few words for bracket: use \left ( \sum x^2 \right)^2 for bigget size bracket..I mean \left ( \right) \[ \left ( \sum x^2 \right)^2\]
ex:
\[\left\{ \int x^2\, dx \right \}^2=1\]
A few words for bracket: use \left ( \sum x^2 \right)^2 for bigget size bracket..I mean \left ( \right) \[ \left ( \sum x^2 \right)^2\]
ex:
\[\left\{ \int x^2\, dx \right \}^2=1\]
"Inspiration is needed in geometry, just as much as in poetry." -- Aleksandr Pushkin
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Please install LaTeX fonts in your PC for better looking equations,
learn how to write equations, and don't forget to read Forum Guide and Rules.
Re: A text book's proof
You can not use $=$ sign there. "$x:x \in ...$' it means $x$ such that $x$ belongs to... . If you write \[A\cup (B \cap C) = x:x\in ... \] it means that set is equal to $x$. You know, that set is not equal to that $x$.
$\Rightarrow$ means 'implies that'
\[A\cup (B \cap C) \Rightarrow x:x\in A\cup (B \cap C)\] এটা ঠিক আছে। $A\cup (B \cap C) $ বলতে এমন সব $x$ কে বুঝায় যেন $x\in A\cup (B \cap C)$
$\Rightarrow$ means 'implies that'
\[A\cup (B \cap C) \Rightarrow x:x\in A\cup (B \cap C)\] এটা ঠিক আছে। $A\cup (B \cap C) $ বলতে এমন সব $x$ কে বুঝায় যেন $x\in A\cup (B \cap C)$
Every logical solution to a problem has its own beauty.
(Important: Please make sure that you have read about the Rules, Posting Permissions and Forum Language)
(Important: Please make sure that you have read about the Rules, Posting Permissions and Forum Language)
Re: A text book's proof
Moon wrote:bracket is a part of LaTeX command...so you have to use \{x\} for $\{x\}$
A few words for bracket: use \left ( \sum x^2 \right)^2 for bigget size bracket..I mean \left ( \right) \[ \left ( \sum x^2 \right)^2\]
ex:
\[\left\{ \int x^2\, dx \right \}^2=1\]
oh, thankS, but, isn't the first bracket a La TeX command??
Last edited by Hasib on Wed Dec 15, 2010 6:27 pm, edited 1 time in total.
A man is not finished when he's defeated, he's finished when he quits.